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Full text of "Mathematical And Physical Papers - Iii"

ON THE MOTION OF PENDULUMS.                          11
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PART I.
ANALYTICAL INVESTIGATION.
SECTION I.
Adaptation of the general equations to the case of the fluid surrounding a lody which oscillates as a pendulum. General laws which follow from the form of the equations. Solution of the equations in the case of an oscillating plane.
1. IN a paper " On the Theories of the Internal Friction of Fluids in Motion, <&c*" which the society did me the honour to publish in the 8th Volume of their Transactions, I have arrived at the following equations for calculating the motion of a fluid when the internal friction of the fluid itself is taken into account, and consequently the pressure not supposed equal in all directions :
dp __    fy    du       du      du       du\        /d*u    d*u    d*u\ -~~~'~W + W + MJ
^3 dx \dx ^ dy     dz
with two more equations which may he written down from symmetry. In these equations u, v, w are the components of the velocity along the rectangular axes of #, y, z* X, Y, Z are the components of the accelerating force ; p is the pressure, t the time, p the density, and //, a certain constant depending on the nature of the fluid.
The three equations of which (1) is the type are not the general equations of motion which apply to a heterogeneous fluid when internal friction is taken into account, which are those numbered 10 in my former paper, but are applicable to a homogeneous incompressible fluid, or to a homogeneous elastic fluid subject to small variations of density, such as those which accompany sonorous vibrations. It must be understood to be included in the term homogeneous that the temperature is uniform throughout
* [Ante, Vol. i. p. 75.]